Improved continued fraction sequence convergent to the Somos' quadratic recurrence constant
type: Item
instance of: scholarly article
title: Improved continued fraction sequence convergent to the Somos' quadratic recurrence constant
zbMATH Open document ID: 1378.11109
author: Xu You
author: Di-Rong Chen
published in: Journal of Mathematical Analysis and Applications
publication date: 2016-01-28T00:00:00Z
review text: The sequence \(g_n=ng^2_{n-1}\) with \(g_0=1\) (defined by Somos) satisfies the asymptotic formula \[ g_n=\sigma^{2^n}\left(n+2-\frac{1}{n}+\frac{4}{n^2}-\frac{21}{n^3}+\cdots\right)^{-1} \] where \(\sigma\) is now known as the Somos' quadratic recurrence constant. Using the fact that \(\sigma=2\exp\{-\frac{1}{2}\gamma(\frac{1}{2})\}\), the authors present an approximation for \(\sigma\) by approximating \(\gamma(\frac{1}{2})\), using the generalized Euler constant function \[ \gamma(z)=\sum\limits_{k=1}^\infty z^{k-1}\left(\frac{1}{k}-\ln\frac{k+1}{k}\right)\quad (|z|\leq 1). \] They also present an approximation for the constant \(\gamma(\frac{1}{3})\).
reviewed by: Anitha Srinivasan
Mathematics Subject Classification ID: 11Y60
Mathematics Subject Classification ID: 11Y65
Mathematics Subject Classification ID: 11A55
Mathematics Subject Classification ID: 41A25
Mathematics Subject Classification ID: 26D15
zbMATH DE Number: 6536920
zbMATH Keywords: Somos' quadratic recurrence constant
zbMATH Keywords: generalized Euler constant
zbMATH Keywords: continued fraction
zbMATH Keywords: multiple-correction method
describes a project that uses: OEIS
MaRDI profile type: Publication
full work available at URL: j.jmaa.2015.12.013
OpenAlex ID: W2295823469
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DOI: 10.1016/J.JMAA.2015.12.013
description: scientific article; zbMATH DE number 6536920